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On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces
DOI:
https://doi.org/10.15446/recolma.v50n1.62187Palabras clave:
Cauchy problem, local and global well-posedness, Benjamin-Ono equation (en)Descargas
We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation ut + uux + βHuxx + (Hux - uxx) = 0, where x ∈ T, t > 0, η > 0 and H denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces Hs(T) for any s > - ½. We also prove some ill-posedness issues when s < -1.
DOI: https://doi.org/10.15446/recolma.v50n1.62187
On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces
Sobre el buen planteamiento de la ecuación de Chen-Lee en espacios de Sobolev periódicos
Ricardo Pastrán1, Oscar Riaño1
1 Universidad Nacional de Colombia, Bogotá, Colombia. rapastranr@unal.edu.co, ogrianoc@unal.edu.co
Abstract
We prove that the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation ut + uux + β H uxx + η (H ux - uxx) = 0, where x ∈ T, t > 0, η > 0 and H denotes the usual Hilbert transform, is locally and globally well-posed in the Sobolev spaces Hs(T) for any s > -½. We also prove some ill-posedness issues when s < -1.
Keywords: Cauchy problem, local and global well-posedness, Benjamin-Ono equation.
2010 Mathematics Subject Classification: 34A12, 35Q35.
Resumen
Probamos que el problema de valor inicial asociado a una perturbación de la ecuación de Benjamín-Ono o ecuación de Chen-Lee ut + uux + β H uxx + η (H ux - uxx) = 0, donde x ∈ T, t > 0, η > 0 y H denota la transformada de Hilbert usual, es localmente y globalmente bien planteado en espacios de Sobolev Hs(T) para cualquier s > -½. También probamos un tipo de mal planteamiento cuando s < -1.
Palabras claves: Problema de Cauchy, buen planteamiento local y global, ecuación de Benjamín-Ono.
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References
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(Recibido: julio de 2015 Aceptado: enero de 2016)
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Derechos de autor 2016 Revista Colombiana de Matemáticas
Esta obra está bajo una licencia internacional Creative Commons Atribución 4.0.